In mathematics a field of sets is a pair where is a set and is an algebra over i.e., a non-empty subset of the power set of closed under the intersection and union of pairs of sets and under complements of individual sets. In other words forms a subalgebra of the power set Boolean algebra of . (Many authors refer to itself as a field of sets. The word "field" in "field of sets" is not used with the meaning of field from field theory.) Elements of are called points and those of are called complexes.
Fields of sets play an essential role in the representation theory of Boolean algebras. Every Boolean algebra can be represented as a field of sets.
Famous quotes containing the words field of, field and/or sets:
“A field of water betrays the spirit that is in the air. It is continually receiving new life and motion from above. It is intermediate in its nature between land and sky.”
—Henry David Thoreau (1817–1862)
“The snow had begun in the gloaming,
And busily all the night
Had been heaping field and highway
With a silence deep and white.”
—James Russell Lowell (1819–1891)
“Wilson adventured for the whole of the human race. Not as a servant, but as a champion. So pure was this motive, so unflecked with anything that his worst enemies could find, except the mildest and most excusable, a personal vanity, practically the minimum to be human, that in a sense his adventure is that of humanity itself. In Wilson, the whole of mankind breaks camp, sets out from home and wrestles with the universe and its gods.”
—William Bolitho (1890–1930)